Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias Samuel Alexander William Alston Anaximander G.E.M.Anscombe Anselm Louise Antony Thomas Aquinas Aristotle David Armstrong Harald Atmanspacher Robert Audi Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer Jeffrey Barrett William Barrett William Belsham Henri Bergson George Berkeley Isaiah Berlin Richard J. Bernstein Bernard Berofsky Robert Bishop Max Black Susanne Bobzien Emil du BoisReymond Hilary Bok Laurence BonJour George Boole Émile Boutroux Daniel Boyd F.H.Bradley C.D.Broad Michael Burke Lawrence Cahoone C.A.Campbell Joseph Keim Campbell Rudolf Carnap Carneades Nancy Cartwright Gregg Caruso Ernst Cassirer David Chalmers Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Antonella Corradini Diodorus Cronus Jonathan Dancy Donald Davidson Mario De Caro Democritus Daniel Dennett Jacques Derrida René Descartes Richard Double Fred Dretske John Dupré John Earman Laura Waddell Ekstrom Epictetus Epicurus Austin Farrer Herbert Feigl Arthur Fine John Martin Fischer Frederic Fitch Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Bas van Fraassen Michael Frede Gottlob Frege Peter Geach Edmund Gettier Carl Ginet Alvin Goldman Gorgias Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie Sam Harris William Hasker R.M.Hare Georg W.F. Hegel Martin Heidegger Heraclitus R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Baron d'Holbach Ted Honderich Pamela Huby David Hume Ferenc Huoranszki Frank Jackson William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan Walter Kaufmann Jaegwon Kim William King Hilary Kornblith Christine Korsgaard Saul Kripke Thomas Kuhn Andrea Lavazza Christoph Lehner Keith Lehrer Gottfried Leibniz Jules Lequyer Leucippus Michael Levin Joseph Levine George Henry Lewes C.I.Lewis David Lewis Peter Lipton C. Lloyd Morgan John Locke Michael Lockwood Arthur O. Lovejoy E. Jonathan Lowe John R. Lucas Lucretius Alasdair MacIntyre Ruth Barcan Marcus Tim Maudlin James Martineau Nicholas Maxwell Storrs McCall Hugh McCann Colin McGinn Michael McKenna Brian McLaughlin John McTaggart Paul E. Meehl Uwe Meixner Alfred Mele Trenton Merricks John Stuart Mill Dickinson Miller G.E.Moore Thomas Nagel Otto Neurath Friedrich Nietzsche John Norton P.H.NowellSmith Robert Nozick William of Ockham Timothy O'Connor Parmenides David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper Porphyry Huw Price H.A.Prichard Protagoras Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Michael Rea Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Richard Rorty Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle JeanPaul Sartre Kenneth Sayre T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Alan Sidelle Ted Sider Henry Sidgwick Walter SinnottArmstrong J.J.C.Smart Saul Smilansky Michael Smith Baruch Spinoza L. Susan Stebbing Isabelle Stengers George F. Stout Galen Strawson Peter Strawson Eleonore Stump Francisco Suárez Richard Taylor Kevin Timpe Mark Twain Peter Unger Peter van Inwagen Manuel Vargas John Venn Kadri Vihvelin Voltaire G.H. von Wright David Foster Wallace R. Jay Wallace W.G.Ward Ted Warfield Roy Weatherford C.F. von Weizsäcker William Whewell Alfred North Whitehead David Widerker David Wiggins Bernard Williams Timothy Williamson Ludwig Wittgenstein Susan Wolf Scientists David Albert Michael Arbib Walter Baade Bernard Baars Jeffrey Bada Leslie Ballentine Marcello Barbieri Gregory Bateson Horace Barlow John S. Bell Mara Beller Charles Bennett Ludwig von Bertalanffy Susan Blackmore Margaret Boden David Bohm Niels Bohr Ludwig Boltzmann Emile Borel Max Born Satyendra Nath Bose Walther Bothe Jean Bricmont Hans Briegel Leon Brillouin Stephen Brush Henry Thomas Buckle S. H. Burbury Melvin Calvin Donald Campbell Sadi Carnot Anthony Cashmore Eric Chaisson Gregory Chaitin JeanPierre Changeux Rudolf Clausius Arthur Holly Compton John Conway Jerry Coyne John Cramer Francis Crick E. P. Culverwell Antonio Damasio Olivier Darrigol Charles Darwin Richard Dawkins Terrence Deacon Lüder Deecke Richard Dedekind Louis de Broglie Stanislas Dehaene Max Delbrück Abraham de Moivre Bernard d'Espagnat Paul Dirac Hans Driesch John Eccles Arthur Stanley Eddington Gerald Edelman Paul Ehrenfest Manfred Eigen Albert Einstein George F. R. Ellis Hugh Everett, III Franz Exner Richard Feynman R. A. Fisher David Foster Joseph Fourier Philipp Frank Steven Frautschi Edward Fredkin Benjamin GalOr Howard Gardner Lila Gatlin Michael Gazzaniga Nicholas GeorgescuRoegen GianCarlo Ghirardi J. Willard Gibbs James J. Gibson Nicolas Gisin Paul Glimcher Thomas Gold A. O. Gomes Brian Goodwin Joshua Greene Dirk ter Haar Jacques Hadamard Mark Hadley Patrick Haggard J. B. S. Haldane Stuart Hameroff Augustin Hamon Sam Harris Ralph Hartley Hyman Hartman Jeff Hawkins JohnDylan Haynes Donald Hebb Martin Heisenberg Werner Heisenberg John Herschel Basil Hiley Art Hobson Jesper Hoffmeyer Don Howard John H. Jackson William Stanley Jevons Roman Jakobson E. T. Jaynes Pascual Jordan Eric Kandel Ruth E. Kastner Stuart Kauffman Martin J. Klein William R. Klemm Christof Koch Simon Kochen Hans Kornhuber Stephen Kosslyn Daniel Koshland Ladislav Kovàč Leopold Kronecker Rolf Landauer Alfred Landé PierreSimon Laplace Karl Lashley David Layzer Joseph LeDoux Gerald Lettvin Gilbert Lewis Benjamin Libet David Lindley Seth Lloyd Hendrik Lorentz Werner Loewenstein Josef Loschmidt Ernst Mach Donald MacKay Henry Margenau Owen Maroney David Marr Humberto Maturana James Clerk Maxwell Ernst Mayr John McCarthy Warren McCulloch N. David Mermin George Miller Stanley Miller Ulrich Mohrhoff Jacques Monod Vernon Mountcastle Emmy Noether Alexander Oparin Abraham Pais Howard Pattee Wolfgang Pauli Massimo Pauri Wilder Penfield Roger Penrose Steven Pinker Colin Pittendrigh Walter Pitts Max Planck Susan Pockett Henri Poincaré Daniel Pollen Ilya Prigogine Hans Primas Henry Quastler Adolphe Quételet Pasco Rakic Nicolas Rashevsky Lord Rayleigh Jürgen Renn Giacomo Rizzolati Emil Roduner Juan Roederer Jerome Rothstein David Ruelle Tilman Sauer Ferdinand de Saussure Jürgen Schmidhuber Erwin Schrödinger Aaron Schurger Sebastian Seung Thomas Sebeok Franco Selleri Claude Shannon Charles Sherrington David Shiang Abner Shimony Herbert Simon Dean Keith Simonton Edmund Sinnott B. F. Skinner Lee Smolin Ray Solomonoff Roger Sperry John Stachel Henry Stapp Tom Stonier Antoine Suarez Leo Szilard Max Tegmark Teilhard de Chardin Libb Thims William Thomson (Kelvin) Richard Tolman Giulio Tononi Peter Tse Alan Turing Francisco Varela Vlatko Vedral Mikhail Volkenstein Heinz von Foerster Richard von Mises John von Neumann Jakob von Uexküll C. S. Unnikrishnan C. H. Waddington John B. Watson Daniel Wegner Steven Weinberg Paul A. Weiss Herman Weyl John Wheeler Wilhelm Wien Norbert Wiener Eugene Wigner E. O. Wilson Günther Witzany Stephen Wolfram H. Dieter Zeh Semir Zeki Ernst Zermelo Wojciech Zurek Konrad Zuse Fritz Zwicky Presentations Biosemiotics Free Will Mental Causation James Symposium 
Albert Einstein  Emission and Absorption of Radiation in Quantum Theory
Sixteen years ago, when Planck created quantum theory by deriving his radiation
formula, he took the following approach. He calculated the mean energy Ē of a
resonator as a function of temperature according to his newly found quantumtheoretic basic principles, and determined from this the radiation density ρ as a
function of frequency ν and temperature. He accomplished this by deriving — based
upon electromagnetic considerations — a relation between radiation density and
resonator energy Ē:
Ē = c^{3} ρ / 8 π ν^{2}
(1)
His derivation was of unparalleled boldness, but found brilliant confirmation. Not only the radiation formula proper and the calculated value of the elementary quantum in it was confirmed, but also the quantumtheoretically calculated value of Ē was confirmed by later investigations on specific heat. In this manner, equation (1), originally found by electromagnetic reasoning, was also confirmed. However, it remained unsatisfactory that the electromagneticmechanical analysis, which led to (1), is incompatible with quantum theory, and it is not surprising that Planck himself and all theoreticians who work on this topic incessantly tried to modify the theory such as to base it on noncontradictory foundations. Since Bohr's theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained. It so appears that the uniformity of the theory must be established such that the electromagnetomechanical considerations, which led Planck to equation (1), are to be replaced by quantumtheoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the following consideration which is attractive both for its simplicity and generality.
§1. PLANCK's Resonator in a Field of Radiation
The behavior of a monochromatic resonator in a field of radiation, according to the classical theory, can be easily understood if one recalls the manner of treatment that was first used in the theory of Brownian movement. Let E be the energy of the resonator at a given moment in time; we ask for the energy after time τ has elapsed. Hereby, τ is assumed to be large compared to the period of oscillation of the resonator, but still so small that the percentage change of E during τ can be treated as infinitely small. Two kinds of change can be distinguished. First the change
Δ_{1}E =  A E_{τ}
effected by emission; and second, the change Δ_{2}E caused by the work done by the electric field on the resonator. This second change increases with the radiation density and has a "chance"dependent value and a "chance"dependent sign. An electromagnetic, statistical consideration yields the meanvalue relation
< Δ_{2}E > =  B ρ τ
The constants A and B can be calculated in known manner. We call Δ_{1}E the energy change due to emitted radiation, Δ_{2}E the energy change due to incident radiation. Since the mean value of E, taken over many resonators, is supposed to be independent of time, there has to be
< E + Δ_{1}E + Δ_{2}E > = Ē
or
Ē =  (B / A) ρ
One obtains relation (1) if one calculates B and A for the monochromatic resonator in the known way with the help of electromagnetism and mechanics. We now want to undertake corresponding considerations, but on a quantumtheoretical basis and without specialized suppositions about the interaction between radiation and those structures which we want to call "molecules."
§2. Quantum Theory and Radiation
We consider a gas of identical molecules that are in static equilibrium with thermal radiation. Let each molecule be able to assume only a discrete sequence Z_{1}, Z_{2}, etc., of states with energy values ε_{1}, ε_{2}, respectively. Then it follows in known manner and in analogy to statistical mechanics, or directly from Boltzmann's principle, or finally from thermodynamic considerations, that the probability W_{n} of state Z_{n} (or the relative number of molecules which were in state Z_{n}) is given by
W_{n} = p_{n} e^{ εn / k T} (2)
where k is the wellknown Boltzmann constant. p_{n} is the statistical "weight" of state Z_{n}, i.e., a constant that is characteristic of the quantum state of the molecule but independent of the gas temperature T. We shall now assume that a molecule can go from state Z_{n} to state Z_{m} by absorbing radiation of the distinct frequency ν = ν_{nm} ; and likewise from state Z_{m} to state Z_{n} by emitting such radiation. The radiation energy involved is ε_{m}  ε_{n}. In general, this is possible for any combination of two indices m and n. With respect to any of these elementary processes there must be a statistical equilibrium in thermal equilibrium. Therefore, we can confine ourselves to a single elementary process belonging to a distinct pair of indices (n,m). At the thermal equilibrium, as many molecules per time unit will change from state Z_{n} to state Z_{m} under absorption of radiation, as molecules will go from state Z_{m} to state Z_{n} with emission of radiation. We shall state simple hypotheses about these transitions, where our guiding principle is the limiting case of classical theory, as it has been briefly outlined above. We shall distinguish here also two types of transitions: a) Emission of Radiation. This will be a transition from state Z_{m} to state Z_{n} with emission of the radiation energy ε_{m}  ε_{n}. This transition will take place without external influence. One can hardly imagine it to be other than similar to radioactive reactions. The number of transitions per time unit will have to be put at
A_{m}^{n} N_{m},
where A_{m}^{n} is a constant that is characteristic of the combination of the states Z_{m} and Z_{n}, and N_{m} is the number of molecules in state Z_{m}.
b) Incidence of Radiation. Incidence is determined by the radiation within which
the molecule resides; let it be proportional to the radiation density ρ of the effective
frequency. In case of the resonator it may cause a loss in energy as well as an
increase in energy; that is, in our case, it may cause a transition Z_{n} → Z_{m} as well as
a transition Z_{m} →
Z_{n}. The number of transitions
B_{n}^{m} N_{n} ρ,
and the number of transitions Z_{m} → Z_{n} is to be expressed as
B_{m}^{n} N_{m} ρ,
where B_{n}^{m}, B_{m}^{n} are constants related to the combination of states Z_{n}, Z_{m}. As a condition for the statistical equilibrium between the reactions Z_{n} → Z_{m} and Z_{m} → Z_{n} one finds, therefore, the equation
A_{m}^{n} N_{m} +
B_{m}^{n} N_{m} ρ =
B_{n}^{m} N_{n} ρ (3)
Equation (2), on the other hand, yields
N_{n} / N_{m} = (p_{n} / p_{m}) e ^{ ( εm  εn ) / k T} (4)
From (3) and (4) follows
A_{m}^{n} p_{m} = ρ ( B_{n}^{m} p_{n} e ^{ ( εm  εn ) / k T}  B_{m}^{n} p_{m} ) (5)
ρ is the radiation density of that frequency which is emitted with the transition
Z_{m} → Z_{n} and is absorbed with Z_{n} → Z_{m}. Our equation shows the relation between
T and ρ at this frequency.
If we postulate that ρ must approach infinity with ever increasing T, then we necessarily have
B_{n}^{m} p_{n} = B_{m}^{n} p_{m}
(6)
Introducing the abbreviation
A_{m}^{n} / B_{m}^{n} p_{m} = α_{mn},
(7)
one finds
ρ = α_{mn} / ( e^{ ( εm  εn ) / k T}  1 )
(5a)
Einstein has derived Planck's blackbody law
This is Planck's relation between ρ and T with the constants left indeterminate.
The constants A_{m}^{n} and B_{m}^{n} could be calculated directly if we possessed a modified
version of electrodynamics and mechanics that is in compliance with the quantum
hypothesis.
Einstein derives Bohr's second postulate
The fact that ρ must be a universal function of T and ν implies that α_{mn} and
ε_{m}  ε_{n} cannot depend upon the specific constitution of the molecule, but only upon
the effective frequency ν. From Wien's law follows furthermore that α_{mn} must be
proportional to the third power, and ε_{m}  ε_{n} to the first power of ν . Consequently,
one has
ε_{m}  ε_{n} = hν
(8)
where hν is a constant.
While the three hypotheses concerning emission and incidence of radiation lead to Planck's radiation formula, I am of course very willing to admit that this does not elevate them to confirmed results. But the simplicity of the hypotheses, the generality with which the analysis can be carried out so effortlessly, and the natural connection to Planck's linear oscillator (as a limiting case of classical electrodynamics and mechanics) seem to make it highly probable that these are basic traits of a future theoretical representation. The postulated statistical law of emission is nothing but Rutherford's law of radioactive decay, and the law expressed by (8), in conjunction with (5a), is identical with the second basic hypothesis in Bohr's theory of spectra — this too speaks in favor of the theory presented here.
§3. Remark on the Photochemical Law of Equivalence
The photochemical law of equivalence falls in line with our train of thoughts in the
following manner. Let there be a gas of such low temperature that the thermal
radiation of frequency ν, which leads from state Z_{m} to state Z_{n}, does not practically
occur.
According to (2) and (5a), the state Z_{m} will be quite rare compared to state Z_{n}, and we shall assume that almost all gas molecules are in state Z_{n}. Aside from the previously considered process Z_{m} → Z_{n}, let the molecule in state Z_{m} also have the capability of another elementary "chemical" process, e.g., monomolecular dissociation. Let us furthermore assume that the reaction rate of this dissociation is large compared to the rate of occurrence of the reaction Z_{m} → Z_{n}. What will happen now if we irradiate the gas with the effective frequency? Under absorption of the radiation energy ε_{m}  ε_{n} = hν, molecules will continually go from state Z_{n} to state Z_{m}. Only a very small fraction of these molecules will return to state Z_{n} by emission or absorption. Most, by far, will suffer chemical dissociation, corresponding to the postulated higher reaction rate of this process. This means that per dissociating molecule, we will practically find that the radiation energy hν has been absorbed, just as the law of equivalence demands. The essence of this interpretation is that molecular dissociation is achieved by the absorption of light via the quantum state Z_{m}, but not directly without this intermediate state. In consequence, one need not distinguish between a chemically effective and a chemically ineffective absorption of radiation. The absorption of light and the chemical process appear as independent processes. For Teachers
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